Question: Factor the following expression: $x^2 + 2x - 63$
Solution: When we factor a polynomial, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + a)(x + b) &=& xx &+& xb + ax &+& ab \\ \\ &=& x^2 &+& {(a + b)}x &+& {ab} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + a)(x + b) }&\hphantom{=}&\hphantom{ xx }&\hphantom{+}&\hphantom{ (a + b)x }&\hphantom{+}& \\ &=& x^2 & +& {2}x& & {-63} \end{eqnarray} $ The coefficient on the $x$ term is $2$ and the constant term is $-63$ , so to reverse the steps above, we need to find two numbers that add up to $2$ and multiply to $-63$ You can try out different factors of $-63$ to see if you can find two that satisfy both conditions. If you're stuck and can't think of any, you can also rewrite the conditions as a system of equations and try solving for $a$ and $b$ $ {a} + {b} = {2}$ $ {a} \times {b} = {-63}$ The two numbers $9$ and $-7$ satisfy both conditions: $ {9} + {-7} = {2} $ $ {9} \times {-7} = {-63} $ So we can factor the expression as: $(x + {9})(x {-7})$